PHYSICAL REVIEW B 70, 205326 (2004)
Ground state of two-dimensional quantum-dot helium in zero magnetic field:
diagonalization, and variational theory
Perturbation,
Orion Ciftja
Department of Physics, Prairie View A&M University, Prairie View, Texas 77446, USA
A. Anil Kumar
Department of Physics and Department of Electrical Engineering, Prairie View A&M University, Prairie View, Texas 77446, USA
(Received 22 March 2004; revised manuscript received 14 May 2004; published 17 November 2004)
We study the ground-state properties of two-dimensional quantum-dot helium in zero external magnetic field
(a system of two interacting electrons in a two-dimensional parabolic confinement potential) by using perturbation and variational theory. We introduce a family of ground-state trial wave functions with one, two, and
three variational parameters. We compare the perturbation and variational energies with numerically exact
diagonalization results and earlier unrestricted Hartree-Fock studies. We find that the three-parameter variational wave function is an excellent representation of the true ground state and argue on how to generalize such
a wave function for larger quantum dots with arbitrary numbers of electrons.
DOI: 10.1103/PhysRevB.70.205326
PACS number(s): 73.21.La, 31.15.Pf
I. INTRODUCTION
Semiconductor quantum dots are generally fabricated by
applying a lateral confining potential to a two-dimensional
(2D) electron system.1–6 In these systems, the electrons move
in a plane in a lateral confinement potential. The relative
strength of electron-electron and electron-confinement interaction can be experimentally tuned over a wide range resulting in a peculiar electronic system with tunable physical
properties. For this reason, quantum dots are sometimes refered to as “artificial atoms.” In a typical quantum dot, all
electrons are tightly bound, except for a few free electrons.
Such artificial atoms are of immense technological value because they form the building units of larger structures. Quantum dots can contain anything from a single electron to a
collection of thousands of electrons and much of the parameters that describe a quantum dot can be precisely controlled
by conventional nanofabrication methods. Quantum dots are
realized in several ways. One example is a quantum dot array
in which the potential well at the interface is populated with
very few electrons. Another example is one where a charge
coupled device array operated under deep depletion may
have only a few electrons in the potential well under the
gate.
The standard theoretical model of a quantum dot includes
the following approximations. First, the motion of the electrons is considered to be exactly 2D. Second, the confining
potential is taken to be parabolic, and third the interaction
between electrons is considered to be a pure Coulomb interaction. In absence of an external magnetic field, the Hamiltonian for N electrons interacting in a quantum dot is
N
Ĥ =
兺
i=1
冋
册
N
p̂2i
e2
m
,
+ ␻2␳2i +
ជ i − ␳ជ j兩
2m 2
i⬎j 兩␳
兺
共1兲
where pជ̂ i = 共p̂ix , p̂iy兲 and ␳ជ i = 共xi , y i兲 are, respectively, the 2D
momentum operator and position of the ith electron, m is the
electron’s mass, ␻ is the angular frequency of the parabolic
1098-0121/2004/70(20)/205326(8)/$22.50
confining potential, and e is the electron’s charge.
Most of the initial theoretical studies on quantum dots
were done in the strong magnetic field’s regime with all electrons being fully spin polarized. The goal was to study the
crossover regime between microscopic quantum dots and
macroscopic 2D electronic systems (2DES) of the fractional
quantum Hall type.7–10 However, in recent years, there has
been a surge of interest in theoretical studies of quantum dots
in zero magnetic field,11–15 with the main focus on the Fermiliquid–Wigner-solid crossover regime, a problem closely related with the nature of metal-insulator transition in 2D.16
Among the variety of theoretical methods used to study
quantum dots we could mention a few such as analyexact
numerical
diagonaltical
calculations,17–20
izations,21–27 quantum Monte Carlo (QMC) methods,28–31 as
well as density functional theory methods.32–36
The simplest quantum dot system consists of two electrons 共N = 2兲 confined in a 2D parabolic potential in zero
magnetic field. The electrons repel each other with a Coulomb potential and the Hamiltonian of the system is
Ĥ共␳ជ 1, ␳ជ 2兲 =
p̂21 m 2 2 p̂22 m 2 2
e2
+ ␻ ␳1 +
+ ␻ ␳2 +
. 共2兲
2m 2
2m 2
兩␳ជ 1 − ␳ជ 2兩
Because of the analogy of the system to that of helium atom
(although, strictly speaking, the quantum dot is 2D, and the
true helium atom is not) this system is called quantum-dot
helium. It is well known that, because of the Coulomb interaction, no exact analytical solutions can be obtained for the
energy spectra of an arbitrary N-electron parabolic quantum
dot in zero magnetic field. However, in some cases, such as
N = 2 quantum-dot helium, exact numerical diagonalizations22 provide useful input.
Among different quantities describing a quantum dot, the
study of the ground-state properties is of fundamental importance, therefore we focus our main interest on the groundstate wave function and the ground-state energy. Even in the
case of 2D quantum-dot helium in zero magnetic field, an
70 205326-1
©2004 The American Physical Society
PHYSICAL REVIEW B 70, 205326 (2004)
O. CIFTJA AND A. A. KUMAR
exact analytic solution of the problem is not available, therefore most of the methods used so far, rely on approximations
or numerical calculations that some time differ considerably
with each other.
So far, the variational treatment of 2D quantum-dot helium in presence of a magnetic field considers a wave function written as the product of a Laughlin-type wave function
with a Jastrow factor
冉
⌿ = J共␳12兲 ⫻ 共z1 − z2兲兩mz兩 exp −
冊
兩z1兩2 + 兩z2兩2
,
2l2
confinement potential with no magnetic field. In Sec. III we
present the results of first order perturbation theory. In Sec.
IV we numerically solve the problem using exact diagonalization techniques. In Sec. V we present the results obtained
with a one-parameter, two-parameter, and a three-parameter
variational wave function, the last one shown to be a very
good representation of the true ground state. A discussion of
results and concluding remarks can be found in Sec. VI.
II. SINGLE-PARTICLE STATES
共3兲
where J共␳12兲 is the Jastrow factor, z j = x j − iy j is the 2D position in complex notation, j = 1 , 2, and 兩mz兩 = 0 , 1 , . . ., is the
angular momentum. In presence of an applied magnetic field
in the z direction Bz we have l = 冑ប / 共m⍀兲 and ⍀2 = ␻2
+ 共␻c / 2兲2, where ␻c = eBz / m is the cyclotron frequency. The
only types of Jastrow factors used in the literature that we are
aware of have the form J共␳ij兲 ⬀ exp关a␳ij / 共1 + b␳ij兲兴, where a
and b are two variational parameters,37 a generalized version
of the above form used by Pederiva et al.,38 or a simpler
version J共␳ij兲 = 1 used by Bolton.29 The common property of
the above Jastrow factors is that they can reproduce the many
body wave function very well under a magnetic field. However, we argue that in the limit of zero magnetic field, none
of the above choices represents the ground state very well.
Relying on the well-known fact29 that, in zero magnetic field,
the ground state has zero angular momentum, we point out
that the wave function of Eq. (3) with the above choices of
Jastrow factors is no longer the very best description of the
true ground state, therefore alternative scenarios need be
considered. At this point we also want to mention that there
are still controversies for the selection of Jastrow factors, as
discussed in Ref. 29, and more recently in Ref. 38. The main
motivation of the present work is to identify and demonstrate
that a variational wave function with a type of Jastrow factor
different from those previously considered in the literature
constitutes the best choice to describe the ground state of this
system.
In this work, we study the ground-state properties of 2D
quantum-dot helium in zero external magnetic field by using
perturbation and variational theory. Although approximative
in nature, the two approaches will serve as a testing ground
to build high-quality trial wave functions for quantum dot
systems with arbitrary number N of electrons.
Given that a good variational wave function itself can be
an excellent approximation to the true exact ground state,
there is always a need to find better, yet simple enough ways
to accurately describe such complicated systems as the quantum dots. After a systematic study of the ground-state properties of 2D quantum-dot helium by using perturbation and
variational theory, we succesfully identify a class of trial
wave functions with three variational parameters that is indeed an excellent representation of the true ground state as
shown from a comparison of variational energies to respective values obtained from an exact numerical diagonalization
calculation.
In Sec. II we briefly introduce the formalism and describe
the single-particle states for an electron in a 2D parabolic
The problem of a single electron in a 2D parabolic confinement potential with no magnetic field is exactly solvable.
In polar coordinates, the Hamiltonian is written as
Ĥ共␳ជ 兲 =
L̂2
p̂␳2
m
+ z 2 + ␻ 2␳ 2 ,
2m 2m␳
2
共4兲
where m is electron’s mass, ␻ is the angular frequency of the
2D parabolic confinement potential, and
p̂␳2 = − ប2
冉 冊
1 ⳵
⳵
⳵
␳
, L̂z = − iប ,
⳵␸
␳ ⳵␳ ⳵␳
共5兲
where ␳ជ = 共x , y兲 and ␸ is the polar angle. The energy eigenvalues are labeled by the quantum numbers n␳ = 0 , 1 , . . ., and
mz = 0 , ± 1 , ± 2 , . . ., and are given by
En␳mz = ប␻共2n␳ + 兩mz兩 + 1兲.
共6兲
The normalized eigenfunctions are
⌽ n␳mz共 ␳ , ␸ 兲 = N n␳mz
eimz␸
冑2␲ 共␣␳兲
兩mz兩 −␣2␳2/2 兩mz兩 2 2
e
Ln 共␣ ␳ 兲,
␳
共7兲
2 2
z兩
where the functions L兩m
n␳ 共␣ ␳ 兲 are associated Laguerre
39
polynomials. The normalization constant Nn␳mz and the parameter ␣ (that has the dimensionality of an inverse length)
are
N n␳mz =
冑
2n␳ ! ␣2
, ␣=
共n␳ + 兩mz兩兲!
冑
m␻
.
ប
共8兲
The ground-state wave function corresponds to n␳ = 0 and
mz = 0 and is not degenerate:
⌽00共␳, ␸兲 =
␣
冑␲ e
−␣2␳2/2
.
共9兲
A complete set of single-particle solutions of the problem
is the product of the orbitals in Eq. (7) and the spin functions
␹sms共␴兲, where ␴ is the spin variable, s = 1 / 2 is the spin quantum number of the electron, and ms = ± 1 / 2 is the spin angular momentum quantum number. In absence of magnetic
field, the energy eigenvalues do not depend on spin.
III. FIRST-ORDER PERTURBATION THEORY
If we were to ignore the Coulomb repulsion in Eq. (2), we
would have a solution to the eigenvalue problem for the
two-electron system, with eigenfunctions given by
⌽n␳ mz 共␳1 , ␸1兲⌽n␳ mz 共␳2 , ␸2兲 and energy eigenvalues given
1
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1
2
2
PHYSICAL REVIEW B 70, 205326 (2004)
GROUND STATE OF TWO-DIMENSIONAL QUANTUM-DOT…
by E = En␳ mz + En␳ mz . In this idealized model, the ground1
1
2
2
state energy is E = 2ប␻. Since the electrons are identical fermions, the total wave function should be antisymmetric under the interchange of space and spin coordinates. Given that
the space part of the wave function is symmetric, a proper
description of the ground state of this idealized model is
given by the normalized wave function
⌿共␳ជ 1, ␳ជ 2兲 = ⌽00共␳1, ␸1兲⌽00共␳2, ␸2兲Xsinglet
␣2 −共␣2/2兲共␳2+␳2兲
1 2 X
e
singlet ,
␲
=
1
冑2 关␹+共␴1兲␹−共␴2兲 − ␹−共␴1兲␹+共␴2兲兴,
共11兲
where ␹±共␴i兲 = ␹1/2,±1/2共␴i兲 denotes the two possible spin
states of the spin variable ␴i for electron i = 1 and 2. We treat
the Coulomb repulsion as a perturbation and, given that the
ground state is nondegenerate, we can then apply perturbation theory for a nondegenerate state. To first order in perturbation theory, the energy correction of the ground state is
given by
冓
⌬E共1兲 = ⌿共␳ជ 1, ␳ជ 2兲
冏
e
2
兩␳ជ 1 − ␳ជ 2兩
冏
冔
⌿共␳ជ 1, ␳ជ 2兲 .
⌬E
共1兲
冉 冊冕 冕
␣
=
␲
d ␳1
2
d ␳ 2e
2
−␣2共␳21+␳22兲
e
2
兩␳ជ 1 − ␳ជ 2兩
. 共13兲
We carry out the integration by using the identity
⬁
1
兩␳ជ 1 − ␳ជ 2兩
=
兺
m =−⬁
z
冕
⬁
0
dkeimz共␸1−␸2兲Jmz共k␳1兲Jmz共k␳2兲, 共14兲
where Jmz共k␳i兲 are Bessel functions. The integration over angular variables, ␸1 and ␸2 is simple, and after straightforward calculations we obtain
⌬E共1兲 = e2␣
冑
␲
.
2
共15兲
The first-order correction of the energy is a positive contribution, since it arises from repulsive interaction, and when
added to the unperturbed result gives the energy to first order
of perturbation theory E共1兲 = 2ប␻ + ⌬E共1兲. This can be written
as
⑀共1兲共␭兲 =
E共1兲
=2+␭
ប␻
冑
ជ = ␳ជ 1 + ␳ជ 2 , P
ជ = pជ + pជ , rជ = ␳ជ − ␳ជ , pជ = pជ 1 − pជ 2 . 共17兲
R
1
2
1
2
2
2
␲
e 2␣
, ␭=
,
2
ប␻
Accordingly, the total mass of the system is M = 2m and the
reduced mass is ␮ = m / 2. In this representation the Hamiltonian of the system decouples and can be written as
ជ ,rជ兲 = Ĥ 共Rជ 兲 + Ĥ 共rជ兲,
Ĥ共R
R
r
共16兲
P̂2 M 2 2
+ ␻R
2M 2
共19兲
p̂2 ␮ 2 2 e2
+ ␻r + .
2␮ 2
r
共20兲
ជ兲 =
ĤR共R
and
Ĥr共rជ兲 =
The eigenfunctions of the c.m. Hamiltonian are
⌽nRM z共R, ␸R兲 = NnRM z
eiM z␸R
冑2␲ 共␣RR兲
2 2
兩M z兩 −␣R
R /2
e
⫻ Ln兩M z兩共␣R2 R2兲,
R
共21兲
where nR = 0 , 1 , . . ., M z = 0 , ± 1 , . . ., and ␣R = 冑M ␻ / ប = 冑2␣.
The c.m. eigenenergies are of the form EnRM z = 共2nR + 兩M z兩
+ 1兲ប␻. If we ignore the Coulomb interactions for the moment, the eigenenergies and eigenfunctions for the relative
motion Hamiltonian are of the same form as Eqs. (6) and (7),
with the replacement of vector ␳ជ with rជ, of quantum number
n␳ with n, and of ␣ with ␣r = ␣ / 冑2, because of the reduced
mass. As in any standard numerical diagonalization technique we need to calculate the matrix elements of the Hamiltonian in the basis 兩nRM z ; nmz典. The only nondiagonal terms
arise from the Coulomb potential, which is diagonal with
respect to nR, M z, and mz, but not n. As a result the most
general nonzero Hamiltonian matrix elements have the form
具nRM z ; n⬘mz兩Ĥ兩nRM z ; nmz典 = 具nRM z兩ĤR兩nRM z典␦n⬘n
+ 具n⬘mz兩Ĥr兩nmz典. For a given nR, M z, and mz, we have
具nRM z兩ĤR兩nRM z典 / 共ប␻兲 = 共2nR + 兩M z兩 + 1兲,
while
= 具n⬘mz兩Ĥr兩nmz典 / 共ប␻兲 is given by
h n⬘n
hn⬘n = 共2n + 兩mz兩 + 1兲␦nn⬘
共1兲
where ⑀ 共␭兲 is a dimensionless energy (measured in units of
ប␻) and ␭ = e2␣ / 共ប␻兲 is a dimensionless interaction parameter that gauges the strength of the Coulomb correlation relative to the confining oscillator energy.
共18兲
ជ 兲 and Ĥ 共rជ兲 are, respectively, the c.m. and relawhere ĤR共R
r
tive motion Hamiltonians given by
共12兲
Since the Coulomb perturbation does not involve the spin,
we only need calculate the following integral:
2 2
We gauge the accuracy of perturbation theory and, later
on, variational approach, by considering exact numerical diagonalization energies as reference. In the case of 2D
quantum-dot helium, the exact numerical diagonalization
technique simplifies if we separate the Hamiltonian into two
parts, one representing the center-of-mass (c.m.) motion and
the other one representing the relative motion. The c.m. and
relative coordinates are defined as
共10兲
where Xsinglet is the spin singlet state (total spin S = 0 and
M s = 0) given by
Xsinglet =
IV. EXACT NUMERICAL DIAGONALIZATION
+
where
205326-3
␭
冑2
冑
n⬘ ! n!
In n兩m 兩 ,
共n⬘ + 兩mz兩兲 ! 共n + 兩mz兩兲! ⬘ z
共22兲
PHYSICAL REVIEW B 70, 205326 (2004)
O. CIFTJA AND A. A. KUMAR
In⬘n兩mz兩 =
冕
⬁
兩m 兩
z兩
dtt兩mz兩e−tLn z 共t兲L兩m
n 共t兲
⬘
0
1
冑t
共23兲
and ␭ = e2␣ / 共ប␻兲.
For any value of the quantum number mz = 0 , ± 1 , . . ., we
build sufficiently large matrices with elements hn⬘n and diagonalize them by using standard numerical methods. For a
given ␭, the smallest of the eigenvalues represents the
ground state energy for the relative motion. With the addition
of the c.m. energy to the ground-state energy of the relative
motion we obtain the exact numerical diagonalization value
of the ground-state energy of 2D quantum dot helium at zero
magnetic field.
B. Two-parameter variational wave function
Let us now consider a second family of variational wave
functions with two variational parameters. The special form
of the Hamiltonian in Eq. (18) allows a product ansatz for
the wave function in a factorized form
ជ ,rជ兲 = ⌽ 共Rជ 兲⌽ 共rជ兲X
⌿共R
R
r
singlet ,
ជ 兲 are the eigenstates of the c.m. Hamiltonian and
where ⌽R共R
⌽r共rជ兲 is a function of the relative coordinate. The groundstate wave function for the c.m. motion (corresponding to the
energy, ប␻) is found from Eq. (21) and is
⌽00共R, ␸R兲 =
V. VARIATIONAL THEORY
A. One-parameter variational wave function
In this section we employ variational theory and the Ritz
variational principle to calculate the ground-state energy of
2D quantum dot helium. This can be done by choosing a trial
variational wave function that depends on a number of parameters, calculating the expectation value of the Hamiltonian with respect to this trial wave function and then minimizing the trial energy with respect to all parameters.
We start by considering a simple normalized oneparameter variational wave function of the form
⌿a共␳ជ 1, ␳ជ 2兲 =
a2 −共a2/2兲共␳2+␳2兲
1 2 X
e
singlet ,
␲
共24兲
where a is a positive variational parameter to be optimized
and Xsinglet is the spin singlet function.
The problem of determining the ground state energy reduces to the evaluation of the integral
Ea =
冕 冕
d 2␳ 1
d2␳2⌿a共␳ជ 1, ␳ជ 2兲 * Ĥ共␳ជ 1, ␳ជ 2兲⌿a共␳ជ 1, ␳ជ 2兲,
共25兲
冉冊 冉冊 冑冉冊
2
+
␣
a
2
+␭
␲ a
,
2 ␣
共26兲
where ␣ = 冑m␻ / ប and ␭ is the dimensionless Coulomb interaction parameter defined in Eq. (16). It is obvious that the
ground-state energy will depend on the parameter ␭. To simplify notation we introduce a new variable t = a / ␣ and write
⑀共t , ␭兲 = Ea共␭兲 / 共ប␻兲 as
1
⑀共t,␭兲 = t + 2 + ␭
t
2
冑
␲
t.
2
冋
冑␲
2 2
e−␣RR /2 .
共29兲
−
册
冉 冊
L̂2
⳵
␮
ប2 1 ⳵
e2
r
+ z 2 + ␻ 2r 2 +
⌽r共rជ兲 = Er⌽r共rជ兲,
2␮ r ⳵r ⳵r
2␮r
2
r
共30兲
where L̂z is the relative coordinate angular momentum operator and Er are the energy eigenvalues for the relative motion.
There are no exact analytic solutions to the above problem
and almost all treatments envolve some form of approximation to derive useful results.
In view of this and in consideration to the fact that our
main interest is on ground-state properties, we write a twoparameter variational wave function as a product of two
terms, one depending on the c.m. coordinate and the other
one depending on the relative motion coordinate. We choose
a normalized two-parameter variational wave function of the
form
ជ ,rជ兲 = ⌽ 共Rជ 兲⌽ 共rជ兲X
⌿␣R,b共R
␣R
b
singlet ,
共31兲
ជ 兲 = ⌽ 共R , ␸ 兲 is given from Eq. (29) and correwhere ⌽␣R共R
00
R
sponds to the exact ground-state wave function for the c.m.
motion. As a result, there is no need to determine an optimal
value for the parameter ␣R already known to be ␣R / ␣ = 冑2.
On the other hand, we write the relative motion trial wave
function as
⌽b共rជ兲 =
共27兲
Starting from the minimum condition d⑀共t , ␭兲 / dt = 0, we
search for the real (positive) root of equation t0 + ␭冑␲ / 8
= 1 / t30, which represents the optimal value of parameter t0
(therefore the optimal variational parameter a0) that minimizes the energy for any given ␭.
␣R
The eigenvalue equation Ĥr共rជ兲⌽r共rជ兲 = Er⌽r共rជ兲 is more complicated and requires the solution of the stationary
Schrödinger equation
where the Hamiltonian is given in Eq. (2) and the trial wave
function is normalized to 1. The variational energy as a function of the parameter a turns out to be
Ea共␭兲
a
=
ប␻
␣
共28兲
b
冑␲ e
−共b2/2兲r2
.
共32兲
The function, ⌽b共rជ兲 is normalized to one, describes the
ground state of the relative motion and need be optimized by
choosing the optimal value of b via standard variational procedures. The function Xsinglet is the spin singlet function.
The expecation value of the two-body Hamiltonian for the
wave function of Eq. (31) can be reduced to the expression
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PHYSICAL REVIEW B 70, 205326 (2004)
GROUND STATE OF TWO-DIMENSIONAL QUANTUM-DOT…
Eb = ប␻ +
冕
d2r⌽b共rជ兲 * Ĥr共rជ兲⌽b共rជ兲,
共33兲
A calculation of the expectation value of the Hamiltonian
with respect to the above three-parameter trial wave function
gives
where the first term comes from the c.m. motion and the
second integral results from the relative motion. After perfoming the integrals we obtain
冉冊 冉冊
b
Eb共␭兲
=1+
ប␻
␣
2
+
1 ␣
4 b
2
+ ␭冑␲
冉冊
b
.
␣
共34兲
E
=
ប␻
+
As previously done, we introduce the dimensionless variables t = b / ␣ and ⑀共t , ␭兲 = Eb共␭兲 / 共ប␻兲 to obtain
⑀共t,␭兲 = 1 + t2 +
⑀=
− B2 f 1共A,B,c兲 +
1
+ ␭冑␲t.
4t2
共35兲
1
A2
+ 2,
2 2A
f 1共A,B,c兲 =
冕
⬁
0
dtte−共A
冉
C. Three-parameter variational wave function
We can further improve the two-parameter variational
wave function and build a better wave function once we note
that the main effect of the Coulomb repulsion is to push the
two electrons further apart. In absence of Coulomb repulsion, the component of the ground-state wave function that
depends on the relative coordinate is expected to be centered
at 兩␳ជ 1 − ␳ជ 2兩 = 0, but in presence of the Coulomb repulsion between electrons, the center of the relative motion wave function would shift towards nonzero relative coordinates 共兩␳ជ 1
− ␳ជ 2兩 ⫽ 0兲. A variational wave function of the form
a2 2
b2
共␳1 + ␳22兲 − 共␳ជ 1 − ␳ជ 2兲2
2
2
⫻ exp关cb兩␳ជ 1 − ␳ជ 2兩兴Xsinglet
册
共36兲
has all the key ingredients to satisfy the above scenario and
contains only three variational prameters to optimize. The
three parameters a, b, and c are considered to be nonnegative. On physical grounds we may expect to have a / ␣
= 1, since the two electron equation can be separated by the
variables R and r and this choice gives the exact lowest c.m.
energy. Although this argument applies to the three parameter wave function (because of its special form), it is not
always correct and should be treated with some care. This is
because 共␳21 + ␳22兲 depends on both R and r, therefore, for an
arbitrary generic wave function, the optimal value of a may
not always be the value that gives the exact lowest c.m.
energy. For the sake of generality and as a numerical check
of the minimization procedure we prefer to consider a as a
variational parameter. (We later verify that the numerical
minimization procedure always gives the anticipated value
for parameter a.) When b = 0, the three-parameter wave function reduces to the one-parameter wave function (that would
give the exact solution of the problem if we were to ignore
the Coulomb repulsion). When b ⬎ 0, we would then expect
the parameter c to play a significant role on improving the
variational energy, because the main effect of the
c-dependent term is to push the pair of electrons further
apart, therefore optimizing their Coulomb correlations.
f 2共A,B,c兲 =
2/2B2兲t2−t2+2ct
冊
⫻
冋冉
1+
A2
2B2
册
冊
2
t2
A2
A2 c
2
+
c
−
2
−
+
,
2B2
B2 t
− 2ct 1 +
冋
共37兲
where A = a / ␣, B = b / ␣, and ␭ = e2␣ / 共ប␻兲. The four functions
f 1,2,3 and f depend on the variables specified on their arguments. In an integral form they are given by
We minimize ⑀共t , ␭兲 with respect to t in order to obtain the
optimal value of parameter t0 (therefore b0) for any given ␭.
⌿a,b,c共␳ជ 1, ␳ជ 2兲 = exp −
f 2共A,B,c兲
+ ␭Bf 3共A,B,c兲
共4B2兲
f共A,B,c兲
冕
⬁
dtt3e−共A
2/2B2兲t2−t2+2ct
,
0
f 3共A,B,c兲 =
冕
⬁
dte−共A
2/2B2兲t2−t2+2ct
,
0
f共A,B,c兲 =
冕
⬁
dtte−共A
2/2B2兲t2−t2+2ct
,
共38兲
0
where t = b兩␳ជ 1 − ␳ជ 2兩 is an auxiliary variable introduced to simplify the calculation of integrals. For any given value of parameter ␭, we minimize the expression in Eq. (37) with respect to the variational parameters A, B, and c and as a result
obtain the best variational energy and the optimal values of
variational parameters, as well.
VI. RESULTS AND DISCUSSION
The optimal values of variational parameters for the one-,
two-, and three-parameter variational wave functions at different ␭’s are given in Table I. The optimal value of parameter A for the three-parameter variational wave function was
always found to be A = a / ␣ = 1, a value that guarantees the
lowest possible c.m. energy.
In Table II we show the ground-state energies
⑀ = E / 共ប␻兲 for given values of ␭ (first column) obtained from
a variety of different methods: first order perturbation theory,
variational theory, exact numerical diagonalization, and unrestricted Hartree-Fock (HF) technique,40 as well. From the
results of Table II we note that, even the energies obtained
from the one-parameter variational wave function are lower
than the first-order perturbation theory values, for all ␭’s under consideration. As expected, we find out that for a wide
range of Coulomb correlation strengths, the ground-state energy obtained from first-order perturbation theory is too high.
The results suggest that perturbation theory is not very accurate and may be used with some reliability only when 0
205326-5
PHYSICAL REVIEW B 70, 205326 (2004)
O. CIFTJA AND A. A. KUMAR
TABLE I. A display of the optimal variational parameters for
different values of ␭ (first column) corresponding to the different
types of variational wave functions for 2D quantum dot helium. For
the one-parameter variational wave function (var. 1), the optimal
values a / ␣ are listed in the second column. For the two-parameter
variational wave function (var. 2), the optimal values b / ␣ are listed
in the third column and ␣R / ␣ = 冑2 irrespective of ␭. For the threeparameter variational wave function (var. 3), the optimal values of
B = b / ␣ and c are listed, respectively, in the fourth and fifth columns, while we always have A = a / ␣ = 1. The parameter ␣
= 冑m␻ / ប has the dimensionality of an inverse length.
Var. 1
Var. 2
Var. 3
␭
a/␣
b/␣
b/␣
c
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
1.000000
0.873542
0.788274
0.726623
0.679582
0.642208
0.611583
0.585880
0.563892
0.544787
0.527975
0.707107
0.557394
0.480537
0.432455
0.398732
0.373335
0.353267
0.336856
0.323085
0.311297
0.301043
0.000000
0.401849
0.497908
0.542433
0.566026
0.579761
0.588185
0.593558
0.597084
0.599450
0.601063
0.00000
1.67676
2.21655
2.57492
2.85052
3.07820
3.27473
3.44916
3.60699
3.75177
3.88599
艋 ␭ ⬍ 1. Comparing the third and fourth columns of Table II,
we see that the two-parameter variational wave function is a
clear improvement upon the one-parameter variational wave
function, however, the data clearly show that the threeTABLE II. Ground-state energies ⑀ = E / 共ប␻兲 of 2D quantum dot
helium as a function of dimensionless Coulomb coupling parameter
␭ = e2␣ / 共ប␻兲 (first column) obtained from first order perturbation
theory (second column), one-parameter variational wave function
(third column), two-parameter variational wave function (fourth
column), three-parameter variational wave function (fifth column)
are compared with exact numerical diagonalization results (sixth
column) and unrestricted Hartree-Fock (HF) results (Ref. 40),
where available (seventh column). The parameter ␣ = 冑m␻ / ប has
the dimensionality of an inverse length.
␭
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
⑀共1兲共␭兲
⑀ (var. 1) ⑀ (var. 2) ⑀ (var. 3) ⑀ (diag.) ⑀ (HF)
2.00000 2.00000
3.25331 3.16838
4.50663 4.20662
5.75994 5.15405
7.01326 6.03404
8.26657 6.86152
9.51988 7.64662
10.7732 8.39658
12.0265 9.11675
13.2798 9.81125
14.5331 10.48330
2.00000
3.10331
4.01702
4.82331
5.55838
6.24164
6.88494
7.49609
8.08061
8.64257
9.18504
2.00000
3.00174
3.72565
4.32576
4.85637
5.34141
5.79354
6.22032
6.62674
7.01626
7.39141
2.00000
3.00097
3.72143
4.31872
4.84780
5.33224
5.78429
6.21129
6.61804
7.00795
7.38351
4.034
5.182
6.107
6.930
7.686
FIG. 1. Plot of the dimensionless ground-state energy ⑀
= E / 共ប␻兲 of 2D quantum dot helium as a function of the dimensionless Coulomb interaction parameter ␭ = e2␣ / 共ប␻兲 for values of
␭ from 0 to 10. The ground-state energies obtained from the threeparameter wave function (solid circles) are practically identical to
exact numerical diagonalization results (crosses).
parameter variational wave function (fifth column) is the best
choice.
One notes that the variational ground state energies obtained from the three-parameter variational wave function
(fifth column) are extremely close to exact numerical diagonalization values (sixth column), and considerably better
than unrestricted Hartree-Fock (HF) calculations (seventh
column). Furthermore, the results in Table II show that this
excellent agreement between the three-parameter variational
wave function energies and numerical diagonalization results
persists for all ␭’s under consideration (even for very large
␭’s, where Coulomb correlation effects are expected to be
very strong). This further indicates the excellent quality of
this variational wave function.
In Fig. 1 we plot the dimensionless ground-state energy
⑀ = E / 共ប␻兲 as a function of the dimensionless Coulomb interaction parameter ␭ = e2␣ / 共ប␻兲 for values of ␭ ranging
from 0 (no Coulomb repulsion) to ␭ = 10. The results represent the values obtained from first-order perturbation theory
(empty square), one-parameter variational theory (filled
square), two-parameter variational theory (empty circle),
three-parameter variational theory (filled circle), and exact
numerical diagonalization results (crosses). The threeparameter variational wave function energies are practically
identical to the exact numerical diagonalization values.
For all ␭’s under consideration, the three-parameter variational wave function energies are sizeably lower than the
respective unrestricted HF values.40 This is explained by noting that, in the absence of a magnetic field, the spin singlet
state is the state of lowest energy, therefore the neglect of
electronic pair correlations in the HF approach is responsible
for the poor quality of the HF singlet ground state. On the
contrary, the three-parameter variational wave function captures all essential electronic correlations through the parameter c-dependent term.
With the intent to generalize such a wave function to
larger quantum dots, we can rewrite the three-parameter
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PHYSICAL REVIEW B 70, 205326 (2004)
GROUND STATE OF TWO-DIMENSIONAL QUANTUM-DOT…
variational wave function for 2D quantum dot helium as
N=2
J共␳ij兲,
兿
i⬍j
⌿ = D ↑D ↓
共39兲
where D↑, D↓ are Slater determinants for spin “up” and spin
“down” states and J共␳ij兲 is a two-body Jastrow correlation
factor which in our case has the form
冉
冊
b2 2
␳ + cb␳ij ,
2 ij
J共␳ij兲 = exp −
共40兲
where ␳ij = 兩␳ជ i − ␳ជ j兩 and b, c are non-negative variational parameters to be optimized. It is obvious that the wave function
in Eq. (39) is equivalent to the wave function in Eq. (36)
since D↑D↓ ⬀ exp关−共a2 / 2兲共␳21 + ␳22兲兴 for N = 2. To the best of
our knowledge, a three-parameter trial wave function with a
Jastrow factor of the form of Eq. (40) has not been previously considered neither for N = 2, nor for arbitrary N quantum dots, with or without magnetic field. Therefore, we believe it is appropriate to compare this trial wave function to
other different trial wave functions considered in the literature.
For example, Harju et al.37 have studied a 2D N = 2 quantum dot in presence of a magnetic field using a variational
wave function of the form of Eq. (3) with Jastrow factor of
the form
冉
J共␳ij兲 ⬀ exp
冊
a␳ij
,
1 + b␳ij
共41兲
where a and b are two variational parameters.
Although the trial wave function of Eq. (36) cannot be
directly compared to that in Eq. (3), since the first describes
2D quantum dot helium at zero magnetic field, while the
second considers a nonzero magnetic field, it is evident that
these two variational wave functions have very different Jastrow factors. Furthermore, as noted in Ref. 37, the Jastrow
ansatz of Eq. (41) does not have the right asymptotic behavior for ␳ij → ⬁, where the electronic correlations are expected
to vanish and not saturate to a finite value. In a separate
study, Bolton29 considered systems of N = 2 − 4 interacting
electrons in a quantum dot, in the presence of magnetic field,
using the fixed-node diffusion Monte Carlo (DMC) method.
For N = 2 electrons and a starting wave function of Laughlin
form, he found out that, without Coulomb interaction, the
state with angular momentum mz = 0 remains the ground state
over the entire range of magnetic fields (nonzero and zero
magnetic field). With the inclusion of the Coulomb interaction, the situation at zero magnetic field is not altered, although this is no longer the case for a nonzero magnetic
field. It is obvious that, for zero magnetic field and angular
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momentum mz = 0, the Laughlin factor of the ground-state
2 2
2
wave function in Eq. (3) reduces to ⬀e−共␣ /2兲共␳1+␳2兲, which
corresponds to the un-normalized product of wave functions
for two uncorrelated electrons. Therefore, in the limit of zero
magnetic field, the wave function of Eq. (3) transforms into a
wave function of the form of Eq. (36) possesing the same
one-body term, but having a very different two-body Jastrow
factor.
We point out that generalizations of trial wave functions
as in Eq. (39) in absence31 and in presence of weak magnetic
fields41 have always considered Jastrow correlation factors of
the type of Eq. (41) so far. A generalized version of the
Jastrow factor in Eq. (41) was also used in recent DMC
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the variational wave functions under consideration in this
work have substantially different Jastrow correlation factors
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We can readily generalize the wave function of Eq. (39)
with the proposed Jastrow factor of Eq. (40) to quantum dots
with arbitrary number N of electrons. By doing so, the product of Jastrow factors is now extended to all pairs of electrons and we need to consider Slater determinants of N↑ spin
“up” and N↓ spin “down” electrons, with N↑ + N↓ = N. As a
first scenario, one might consider the variational parameters,
b and c to be the same for any pair of electrons, irrespective
of their relative spin. Under a more sophisticated treatment,
one would assume different variational parameters b and c
for pairs of electrons with parallel and opposite spins. This is
because in larger systems we have correlations between pairs
of electrons not only of opposite spin (as in the case of N
= 2 quantum-dot helium), but also of pairs of electrons of
parallel spins. It is reasonable to expect that pairs of electrons with parallel and opposite spins will not correlate in the
same way, under the most general circumstances.
Whether we consider the first or the second scenario (with
more variational parameters), standard variational Monte
Carlo (VMC) techniques would be applied to get an accurate
evaluation of the relevant physical properties of the system.
Trial wave functions of this type could also be good starting
points to implement additional DMC simulations as those
performed by Bolton29 for 2D quantum dots with few electrons. Work in this direction is in progress.
ACKNOWLEDGMENTS
Part of this work was supported by the Office of the VicePresident for Research and Development of Prairie View
A&M University through a 2003-2004 Research Enhancement Program grant.
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